1. Field of the Invention
The present invention relates generally to the estimation of spectral moments of weather radar returns and, more particularly, to an apparatus and method for estimating hydrometeor reflectivity, mean Doppler velocity and Doppler velocity spectral width.
2. Description of the Prior Art
Pulse Doppler radar is used to map severe storm reflectivity and velocity structure. The radar beam's ability to penetrate thunderstorms and clouds permits the determination of the dynamic structure of otherwise unobservable events. This inside look has enabled meteorologists to understand the life cycle and dynamics of storms as well as provide timely severe storm warnings.
Real-time reflectivity PPI maps have been available to meteorologists since the mid-1940's. Real-time Doppler velocity maps were not available until the late 1960's. Since that time meteorologists have been pre-occupied with the development of systems that provide real-time spectral measurements in every radar volume within the radar beam, of which mean Doppler velocity and Doppler velocity spectral width are of greatest interest. Thus, the basic task of a modern weather Doppler radar is to provide estimates of the first three spectral moments of weather radar returns. The zero moment is a measure of average signal power return from which "reflectivity" can be calculated. This "reflectivity" is related to the total water content of the volume illuminated by the radar signal. The first moment about zero or "mean Doppler velocity" is a measure of the average radial velocity of hydrometeors in the sample volume. The square root of the second moment about the spectral mean or "Doppler velocity spectral width" is a measure of hydrometeor radial velocity dispersion.
Weather spectral moment estimates derived from the radar returns are uncertain, at any point in time, even for large signal-to-noise (S/N) ratios. This uncertainty is due to the fact that weather radar returns exhibit many of the characteristics of a non-stationary random process and only a finite number of radar data samples are available for each sampled radar volume upon which to base these estimates. These estimates are further degraded at low S/N due to additive white Gaussian noise.
Mean radar power return and reflectivity are related by the radar weather equation: ##EQU1## where: P=mean received signal power (milli-watts)
P.sub.t =peak transmitted power (watts) PA1 g=one way antenna gain PA1 l=one way propagation attenuation PA1 l.sub.r =radar losses PA1 .gamma.=pulse width (microseconds) PA1 .theta.=one way 3 dB width of circular antenna pattern (rad) PA1 k.sub.w =complex refractive index of water PA1 Z.sub.e =effective reflectivity factor (mm.sup.6 /m.sup.3) PA1 r.sub.o =range (km) PA1 .lambda.=radar wavelength (cm) PA1 .sub.d =estimated mean Doppler frequency (rad/s) PA1 .lambda.=radar wavelength (m) PA1 T=radar pulse repetition interval (s) ##EQU3## (k=0, . . . , M-1) M=number of received pulses PA1 I.sub.i =in-phase video sample of ith received pulse PA1 Q.sub.i =quadrature video sample of ith received pulse PA1 * denotes complex conjugate PA1 1) The phase of arg{R(kT)} is unwrapped for lags K=1, . . . , 2 p using a fast Fourier transform (FFT) based estimate of f.sub.d to calculate a reference phase value at each lag. PA1 2) A weighted least mean square error fit is made to the phase unwrapped arg{R(kT)} for lags k=1,2, . . . ,2 p, by a third degree odd polynomial where errors are weighted by W(kT)=exp{-(kT).sup.2 /2(.sigma..sub.t).sup.2 }, and .sigma..sub.t is derived from the first processor's estimate of .sigma..sub.f. PA1 3) The coefficient of the polynomial's linear term is selected as an unbiased estimate of w.sub.d.
Mean Doppler radial velocity may be estimated in the time domain from the argument of the received signal autocorrelation function as follows: ##EQU2## where V=estimated mean radial velocity (m/s)
z=sample of complex video return (z.sub.i =I.sub.i +jQ.sub.i)
Estimates v and w.sub.d are unbiased when the received signal power spectral density is symmetric. This technique, dubbed pulse-pair processing, processes contiguous, overlapping, consecutive pairs of correlated returns from the same target volume.
Theoretical justification for pulse-pair processing rests on the fact that it has been shown that pulse-pair processing of non-overlapping, statistically independent, pulse-pairs is a maximum likelihood estimator of mean velocity of a narrow band Gaussian signal process, with an arbitrary shaped spectrum, that is immersed in Gaussian noise having an arbitrary shaped power spectral density.
Doppler velocity spectral width .sigma..sub.v may also be calculated in the time domain from the magnitude of the received signal autocorrelation function at several lags, assuming a Gaussian shaped signal power spectral density. Several well known expressions for estimating Doppler frequency spectral width .sigma..sub.f which is related to Doppler velocity spectral width .sigma..sub.v by .sigma..sub.f =2.sigma..sub.v/.lambda. are: ##EQU4## where N denotes received noise power.
It is well known that maximum likelihood estimates of spectral moments are quite complicated, difficult to compute, and closed form solutions usually do not exist. The maximum likelihood estimate of mean velocity v for a Gaussian signal process, with a Gaussian shaped power spectral density, immersed in additive white Gaussian noise may be expressed as; ##EQU5## where C.sub.k are derived from maximum likelihood estimates of the elements of the inverse covariance matrix of the received signal-plus-noise time series. Although the C.sub.k do not depend on v, they do depend on estimates of the received signal power and spectral width. Hence the estimate of v is coupled to the other two moments. Further, the above expression for v is implicit since v appears on both sides of a transcendental equation. An iterative technique involving, an inordinate number of steps, is required to solve for v. For example, for 16 received pulses (M=16) 137 coupled nonlinear equations must be solved.
The mathematical structure of the maximum likelihood estimator that operates on contiguous pulse-pairs suggested that simpler sub-optimum approaches might be found which used higher order lag values of the autocorrelation function to improve spectral mean accuracy at low signal-noise-ratios (S/N), where estimates of R(kT) are statistically independent for different k. A poly-pulse-pair estimator has been proposed for processing contiguous overlapping pulse-pairs which averages independent higher order lag spectral means estimates to provide v.sub.k, the estimate obtained from lag k, v.sub.k having the following form: ##EQU6## Estimates of v.sub.k may be weighted prior to averaging, either to minimize the error in the estimate of v.sub.k, or more simply, to weight each v.sub.k uniformly. This poly-pulse-pair estimator also requires the argument of R(kT) to be phase unwrapped since higher order lag phase values typically exceed 2.pi..
Monte Carlo simulation of poly-pulse-pair processing, with equal weighting of v.sub.k, has been shown to be more accurate than pulse-pair processing at low S/N but less accurate at high S/N. Since combining multiple estimates of v.sub.k theoretically contains more information about v than that in v.sub.1, the latter result was unanticipated and has not been adequately explained.
Few explicit maximum likelihood spectral estimators are known in the prior art. Further, maximum likelihood estimates of the first three spectral moments are in general mathematically coupled. Prior art efforts have been directed at finding sub-optimum uncoupled estimators having error variances that approach the Cramer-Rao lower bound. Because of computational simplicity, pulse-pair processing is widely used for estimating mean Doppler velocity. Poly-pulse-pair processing, as presently configured, yields more accurate estimates of the first spectral moment than pulse-pair processing at low S/N but not at high S/N.